[Paper Review] The Denominators of Power Sums of Arithmetic Progressions

Bernd C. Kellner, Jonathan Sondow|arXiv (Cornell University)|May 15, 2017

Analytic Number Theory Research6 references3 citations

TL;DR

This paper derives an explicit formula for the denominator of power sums of arithmetic progressions, extending Bernoulli's classical formula. It establishes a criterion for integrality of polynomial coefficients and characterizes when ratios of successive denominators of Bernoulli polynomials are infinitely often integers, using base-$p$ digit sums and properties of radical and least common multiples of denominators.

ABSTRACT

In a recent paper the authors studied the denominators of polynomials that represent power sums by Bernoulli's formula. Here we extend our results to power sums of arithmetic progressions. In particular, we obtain a simple explicit criterion for integrality of the coefficients of these polynomials. As applications, we obtain new results on the sequence of denominators of the Bernoulli polynomials. A consequence is that certain quotients of successive denominators are infinitely often integers, which we characterize.

Motivation & Objective

• To extend the known denominator formula for classical power sums to power sums over arithmetic progressions.
• To derive a criterion for when the coefficients of these generalized power sum polynomials are integral.
• To characterize the conditions under which successive denominators of Bernoulli polynomials yield integer ratios, infinitely often.
• To clarify the role of base-$p$ digit sums and radical functions in determining the structure of these denominators.

Proposed method

• Derives an explicit formula for the denominator of $\mathscr{S}_{m,r}^{n}(x)$, the power sum of an arithmetic progression, using the generalized Bernoulli formula.
• Applies the classical denominator formula for $S_n(x)$ to the generalized case via transformation of the Bernoulli polynomial argument.
• Uses the sum of base-$p$ digits $s_p(n)$ to characterize the prime factors of the denominator $\mathbb{D}_n = \operatorname{denom}(B_n(x) - B_n)$.
• Introduces the radical function $\operatorname{rad}(k)$ and least common multiples to express $\mathfrak{D}_n = \operatorname{denom}(B_n(x))$ and $\mathbf{D}_n = \operatorname{denom}(B_n)$.
• Employs asymptotic bounds on digit sums $s_p(q^k)$ to prove that $s_p(q^k) \to \infty$ as $k \to \infty$, enabling the characterization of infinite integer ratios.
• Leverages results from Almkvist and Meurman on integer-valued Bernoulli polynomial evaluations at rational points to support the main theorems.

Experimental results

Research questions

• RQ1 Under what conditions is the polynomial $\mathscr{S}_{m,r}^{n}(x)$ with coefficients in $\mathbb{Z}[x]$?
• RQ2 When are the ratios $\mathbb{D}_n / \mathbb{D}_{n+1}$ and $\mathfrak{D}_n / \mathfrak{D}_{n+1}$ infinitely often integers?
• RQ3 How do the base-$p$ digit sums $s_p(n)$ and the radical function $\operatorname{rad}(n+1)$ determine the structure of the denominators of Bernoulli polynomials?
• RQ4 What is the precise formula for the denominator of the power sum $\mathscr{S}_{m,r}^{n}(x)$ over an arithmetic progression with parameters $m$ and $r$?
• RQ5 How does the independence of $\operatorname{denom}(\mathscr{S}_{m,r}^{n}(x))$ on $r$ arise from the underlying algebraic structure?

Key findings

• The denominator of $\mathscr{S}_{m,r}^{n}(x)$ is given by $\frac{n+1}{\gcd(n+1,m^n)} \cdot \frac{\mathbb{D}_{n+1}}{\gcd(\mathbb{D}_{n+1},m)}$, and is independent of $r$.
• The polynomial $\mathscr{S}_{m,r}^{n}(x)$ lies in $\mathbb{Z}[x]$ if and only if $\mathfrak{D}_n \mid m$, where $\mathfrak{D}_n = \operatorname{lcm}(\mathbb{D}_n, \mathbf{D}_n)$.
• The ratio $\mathbb{D}_n / \mathbb{D}_{n+1}$ is infinitely often an integer, and the paper characterizes the cases where it equals 1, 2, or an odd prime $p$.
• For $n = 2^k - 1$, the ratio $\mathbb{D}_n / \mathbb{D}_{n+1} = 2$ if $n+1$ is not a power of 2, and equals 1 or $p$ otherwise depending on the base-$p$ digit sum of $n+1$.
• When $n = p^k - 1$ for an odd prime $p$, the ratio $\mathfrak{D}_n / \mathfrak{D}_{n+1} = p$, and for $n = p^k q^\ell - 1$, the ratio is $1$, $p$, $q$, or $pq$ depending on the digit sums $s_p(q^\ell)$ and $s_q(p^k)$.
• The ratio $\mathfrak{D}_n / \mathfrak{D}_{n+1}$ is always odd when $n \geq 2$ is even, and the structure of the denominator is governed by the squarefree and even nature of $\mathfrak{D}_n$.